Theorem bj-ceqsalt 36847

Description: Remove from ceqsalt 3478 dependency on ax-ext 2701 (and on df-cleq 2721 and df-v 3446). Note: this is not doable with ceqsralt 3479 (or ceqsralv 3485), which uses eleq1 2816, but the same dependence removal is possible for ceqsalg 3480, ceqsal 3482, ceqsalv 3484, cgsexg 3489, cgsex2g 3490, cgsex4g 3491, ceqsex 3493, ceqsexv 3495, ceqsex2 3498, ceqsex2v 3499, ceqsex3v 3500, ceqsex4v 3501, ceqsex6v 3502, ceqsex8v 3503, gencbvex 3504 (after changing 𝐴 = 𝑦 to 𝑦 = 𝐴), gencbvex2 3505, gencbval 3506, vtoclgft 3515 (it uses Ⅎ, whose justification nfcjust 2877 does not use ax-ext 2701) and several other vtocl* theorems (see for instance bj-vtoclg1f 36879). See also bj-ceqsaltv 36848. (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 2810	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2	1	3anim3i 1154	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥 𝑥 = 𝐴))
3		bj-ceqsalt0 36845	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥 𝑥 = 𝐴) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
4	2, 3	syl 17	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086  ∀wal 1538   = wceq 1540  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-clel 2803

This theorem is referenced by:  bj-ceqsalgALT  36851