Theorem lemma-L6.02a 726

Description: Simular to solvesub-P6a 704, but using ax-L12 29.

'𝑥' cannot occur in '𝑡'.

Hypotheses

Ref	Expression
lemma-L6.02a.1	⊢ Ⅎ𝑥𝜓
lemma-L6.02a.2	⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
lemma-L6.02a	⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ 𝜓)

Distinct variable group:   𝑡,𝑥

Proof of Theorem lemma-L6.02a

Step	Hyp	Ref	Expression
1		lemma-L6.02a.2	. . . 4 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))
2		imoverbi-P4.30b 479	. . . 4 ⊢ ((𝑥 = 𝑡 → (𝜑 ↔ 𝜓)) ↔ ((𝑥 = 𝑡 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜓)))
3	1, 2	bimpf-P4.RC 532	. . 3 ⊢ ((𝑥 = 𝑡 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜓))
4	3	suballinf-P5 594	. 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜓))
5		lemma-L6.02a.1	. . 3 ⊢ Ⅎ𝑥𝜓
6	5	qimeqallb-P6 701	. 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜓) ↔ (∃𝑥 𝑥 = 𝑡 → ∀𝑥𝜓))
7		axL6ex-P5 625	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑡
8	7	thmeqtrue-P4.21a 442	. . 3 ⊢ (∃𝑥 𝑥 = 𝑡 ↔ ⊤)
9	5	qremall-P6 722	. . 3 ⊢ (∀𝑥𝜓 ↔ 𝜓)
10	8, 9	subimd-P3.40c.RC 330	. 2 ⊢ ((∃𝑥 𝑥 = 𝑡 → ∀𝑥𝜓) ↔ (⊤ → 𝜓))
11		trueie-P4.22a 444	. 2 ⊢ ((⊤ → 𝜓) ↔ 𝜓)
12	4, 6, 10, 11	tbitrns-P4.17.RC 431	1 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ 𝜓)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  ⊤wff-true 153  ∃wff-exists 595  Ⅎwff-nfree 681

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L12 29

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682

This theorem is referenced by:  isubtopsubv-P6  727  lemma-L6.03a  728