Theorem lemma-L5.02a 653

Description: A lemma used for bound variable substitution and specialization theorems.

'𝑥' cannot occur in either '𝑡' or '𝜓'.

The '→' in the hypothesis is replaced with a '↔' in specisub-P5 654 to make the practical usage more clear.

Hypothesis

Ref	Expression
lemma-L5.02a.1	⊢ (𝑥 = 𝑡 → (𝜑 → 𝜓))

Assertion

Ref	Expression
lemma-L5.02a	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable groups:   𝜓,𝑥   𝑡,𝑥

Proof of Theorem lemma-L5.02a

Step	Hyp	Ref	Expression
1		lemma-L5.02a.1	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝜑 → 𝜓))
2	1	imcomm-P3.27.RC 266	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑡 → 𝜓))
3	2	dalloverimex-P5.RC.GEN 607	. . 3 ⊢ (∀𝑥𝜑 → (∃𝑥 𝑥 = 𝑡 → ∃𝑥𝜓))
4		axL6ex-P5 625	. . 3 ⊢ ∃𝑥 𝑥 = 𝑡
5	3, 4	mae-P3.23.RC 258	. 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
6		axL5ex-P5 613	. 2 ⊢ (∃𝑥𝜓 → 𝜓)
7	5, 6	syl-P3.24.RC 260	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10  ∃wff-exists 595

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

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

This theorem is referenced by:  specisub-P5  654  cbvallv-P5-L1  658  specw-P5  661