Theorem spcdvw 42224

Description: A version of spcdv 3264 where 𝜓 and 𝜒 are direct substitutions of each other. This theorem is useful because it does not require 𝜑 and 𝑥 to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.)

Hypotheses

Ref	Expression
spcdvw.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
spcdvw.2	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
spcdvw	⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Distinct variable groups:   𝑥,𝐴   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem spcdvw

Step	Hyp	Ref	Expression
1		spcdvw.2	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
2	1	biimpd 218	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜒))
3	2	ax-gen 1713	. . 3 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒))
4	3	a1i 11	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)))
5		spcdvw.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
6		nfv 1830	. . 3 ⊢ Ⅎ𝑥𝜒
7		nfcv 2751	. . 3 ⊢ Ⅎ𝑥𝐴
8	6, 7	spcimgft 3257	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜓 → 𝜒)))
9	4, 5, 8	sylc 63	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175

This theorem is referenced by:  setrec1lem4  42236