Theorem spcdvw 50067

Description: A version of spcdv 3550 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	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
2	1	biimpd 229	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜒))
3	2	ax-gen 1797	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒))
4		spcdvw.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
5		nfv 1916	. . 3 ⊢ Ⅎ𝑥𝜒
6		nfcv 2899	. . 3 ⊢ Ⅎ𝑥𝐴
7	5, 6	spcimgfi1 3506	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜓 → 𝜒)))
8	3, 4, 7	mpsyl 68	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542   ∈ wcel 2114

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-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-cleq 2729  df-clel 2812  df-nfc 2886

This theorem is referenced by:  setrec1lem4  50078