Theorem dral1-o 37769

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 2438 using ax-c11 37752. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)

Hypothesis

Ref	Expression
dral1-o.1	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
dral1-o	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Proof of Theorem dral1-o

Step	Hyp	Ref	Expression
1		hbae-o 37768	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∀𝑥 𝑥 = 𝑦)
2		dral1-o.1	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	biimpd 228	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → 𝜓))
4	1, 3	alimdh 1819	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑥𝜓))
5		ax-c11 37752	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓))
6	4, 5	syld 47	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜓))
7		hbae-o 37768	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦)
8	2	biimprd 247	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜓 → 𝜑))
9	7, 8	alimdh 1819	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑦𝜑))
10		ax-c11 37752	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦𝜑 → ∀𝑥𝜑))
11	10	aecoms-o 37767	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
12	9, 11	syld 47	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜑))
13	6, 12	impbid 211	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-11 2154  ax-c5 37748  ax-c4 37749  ax-c7 37750  ax-c10 37751  ax-c11 37752  ax-c9 37755

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782

This theorem is referenced by:  ax12fromc15  37770  axc16g-o  37799  ax12indalem  37810  ax12inda2ALT  37811