Theorem bj-ceqsalt 37975

Description: Remove from ceqsalt 3487 dependency on ax-ext 2735 (and on df-cleq 2755 and df-v 3456). Note: this is not doable with ceqsralt 3488 (or ceqsralv 3494), which uses eleq1 2852, but the same dependence removal is possible for ceqsalg 3489, ceqsal 3491, ceqsalv 3493, cgsexg 3498, cgsex2g 3499, cgsex4g 3500, ceqsex 3501, ceqsexv 3502, ceqsex2 3504, ceqsex2v 3505, ceqsex3v 3506, ceqsex4v 3507, ceqsex6v 3508, ceqsex8v 3509, gencbvex 3510 (after changing 𝐴 = 𝑦 to 𝑦 = 𝐴), gencbvex2 3511, gencbval 3512, vtoclgft 3519 (it uses Ⅎ, whose justification nfcjust 2912 does not use ax-ext 2735) and several other vtocl* theorems (see for instance bj-vtoclg1f 38007). See also bj-ceqsaltv 37976. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ceqsalt	⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem bj-ceqsalt

Step	Hyp	Ref	Expression
1		elisset 2846	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2	1	3anim3i 1172	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥 𝑥 = 𝐴))
3		bj-ceqsalt0 37973	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥 𝑥 = 𝐴) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
4	2, 3	syl 18	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812  Ⅎwnf 1816   ∈ wcel 2147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-clel 2839

This theorem is used by:  bj-ceqsalgALT  37979