Theorem spcdvw 50305

Description: A version of spcdv 3555 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 231	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜒))
3	2	ax-gen 1817	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒))
4		spcdvw.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
5		nfv 1936	. . 3 ⊢ Ⅎ𝑥𝜒
6		nfcv 2926	. . 3 ⊢ Ⅎ𝑥𝐴
7	5, 6	spcimgfi1 3517	. 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 208  ∀wal 1560   = wceq 1562   ∈ wcel 2144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1802  df-nf 1806  df-cleq 2756  df-clel 2839  df-nfc 2913

This theorem is referenced by:  setrec1lem4  50316