Theorem sbceq1d 3056

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 3053	. 2 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))
3	1, 2	syl 14	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   = wceq 1402  [wsbc 3051

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-sbc 3052

This theorem is used by:  sbceq1dd  3057  rexrnmpt  5851  findcard2  7193  findcard2s  7194  ac6sfi  7202  nn1suc  9325  uzind4s  9999  uzind4s2  10000  fzrevral  10522  fzshftral  10525  wrdind  11508  wrd2ind  11509  cjth  11625  prmind2  12914  issrg  14318  islmod  14676  isassa  15051  bj-bdfindes  17073  bj-findes  17105