Theorem sbceq1d 3037

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 3034	. 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 105   = wceq 1398  [wsbc 3032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227  df-sbc 3033

This theorem is referenced by:  sbceq1dd  3038  rexrnmpt  5798  findcard2  7121  findcard2s  7122  ac6sfi  7130  nn1suc  9204  uzind4s  9868  uzind4s2  9869  fzrevral  10385  fzshftral  10388  wrdind  11352  wrd2ind  11353  cjth  11469  prmind2  12755  issrg  14042  islmod  14370  bj-bdfindes  16648  bj-findes  16680