Theorem sbcbi1 3786

Description: Distribution of class substitution over biconditional. One direction of sbcbig 3780 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Assertion

Ref	Expression
sbcbi1	⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))

Proof of Theorem sbcbi1

Step	Hyp	Ref	Expression
1		sbcex 3738	. 2 ⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → 𝐴 ∈ V)
2		sbcbig 3780	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))
3	2	biimpd 229	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))
4	1, 3	mpcom 38	1 ⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114  Vcvv 3429  [wsbc 3728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-sbc 3729

This theorem is referenced by:  dfconngr1  30258