Theorem gencl 2586

Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Hypotheses

Ref	Expression
gencl.1	⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝐵))
gencl.2	⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))
gencl.3	⊢ (𝜒 → 𝜑)

Assertion

Ref	Expression
gencl	⊢ (𝜃 → 𝜓)

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)   𝜃(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem gencl

Step	Hyp	Ref	Expression
1		gencl.1	. 2 ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝐵))
2		gencl.3	. . . . 5 ⊢ (𝜒 → 𝜑)
3		gencl.2	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))
4	2, 3	syl5ib 143	. . . 4 ⊢ (𝐴 = 𝐵 → (𝜒 → 𝜓))
5	4	impcom 116	. . 3 ⊢ ((𝜒 ∧ 𝐴 = 𝐵) → 𝜓)
6	5	exlimiv 1489	. 2 ⊢ (∃𝑥(𝜒 ∧ 𝐴 = 𝐵) → 𝜓)
7	1, 6	sylbi 114	1 ⊢ (𝜃 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-gen 1338  ax-ie2 1383  ax-17 1419

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  2gencl  2587  3gencl  2588  axprecex  6954  axpre-ltirr  6956