Theorem sbceq1d 2769

Description: Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)

Hypothesis

Ref	Expression
sbceq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
sbceq1d	⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))

Proof of Theorem sbceq1d

Step	Hyp	Ref	Expression
1		sbceq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		dfsbcq 2766	. 2 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))
3	1, 2	syl 14	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243  [wsbc 2764

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-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-sbc 2765

This theorem is referenced by:  sbceq1dd  2770  rexrnmpt  5310  findcard2  6346  findcard2s  6347  ac6sfi  6352  nn1suc  7933  uzind4s  8533  uzind4s2  8534  fzrevral  8967  fzshftral  8970  cjth  9446  bj-bdfindes  10074  bj-findes  10106