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Theorem reusv3i 4137
Description: Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
Hypotheses
Ref Expression
reusv3.1 (y = z → (φψ))
reusv3.2 (y = z𝐶 = 𝐷)
Assertion
Ref Expression
reusv3i (x A y B (φx = 𝐶) → y B z B ((φ ψ) → 𝐶 = 𝐷))
Distinct variable groups:   x,y,z,B   x,𝐶,z   x,𝐷,y   φ,x,z   ψ,x,y
Allowed substitution hints:   φ(y)   ψ(z)   A(x,y,z)   𝐶(y)   𝐷(z)

Proof of Theorem reusv3i
StepHypRef Expression
1 reusv3.1 . . . . . 6 (y = z → (φψ))
2 reusv3.2 . . . . . . 7 (y = z𝐶 = 𝐷)
32eqeq2d 2029 . . . . . 6 (y = z → (x = 𝐶x = 𝐷))
41, 3imbi12d 223 . . . . 5 (y = z → ((φx = 𝐶) ↔ (ψx = 𝐷)))
54cbvralv 2507 . . . 4 (y B (φx = 𝐶) ↔ z B (ψx = 𝐷))
65biimpi 113 . . 3 (y B (φx = 𝐶) → z B (ψx = 𝐷))
7 raaanv 3303 . . . 4 (y B z B ((φx = 𝐶) (ψx = 𝐷)) ↔ (y B (φx = 𝐶) z B (ψx = 𝐷)))
8 prth 326 . . . . . . 7 (((φx = 𝐶) (ψx = 𝐷)) → ((φ ψ) → (x = 𝐶 x = 𝐷)))
9 eqtr2 2036 . . . . . . 7 ((x = 𝐶 x = 𝐷) → 𝐶 = 𝐷)
108, 9syl6 29 . . . . . 6 (((φx = 𝐶) (ψx = 𝐷)) → ((φ ψ) → 𝐶 = 𝐷))
1110ralimi 2358 . . . . 5 (z B ((φx = 𝐶) (ψx = 𝐷)) → z B ((φ ψ) → 𝐶 = 𝐷))
1211ralimi 2358 . . . 4 (y B z B ((φx = 𝐶) (ψx = 𝐷)) → y B z B ((φ ψ) → 𝐶 = 𝐷))
137, 12sylbir 125 . . 3 ((y B (φx = 𝐶) z B (ψx = 𝐷)) → y B z B ((φ ψ) → 𝐶 = 𝐷))
146, 13mpdan 400 . 2 (y B (φx = 𝐶) → y B z B ((φ ψ) → 𝐶 = 𝐷))
1514rexlimivw 2403 1 (x A y B (φx = 𝐶) → y B z B ((φ ψ) → 𝐶 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1226  wral 2280  wrex 2281
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286
This theorem is referenced by:  reusv3  4138
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