Theorem sbceq1ddi 38505

Description: A lemma for eliminating inequality, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)

Hypotheses

Ref	Expression
sbceq1ddi.1	⊢ (𝜑 → 𝐴 = 𝐵)
sbceq1ddi.2	⊢ (𝜓 → 𝜃)
sbceq1ddi.3	⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜃)
sbceq1ddi.4	⊢ ([𝐵 / 𝑥]𝜒 ↔ 𝜂)

Assertion

Ref	Expression
sbceq1ddi	⊢ ((𝜑 ∧ 𝜓) → 𝜂)

Proof of Theorem sbceq1ddi

Step	Hyp	Ref	Expression
1		sbceq1ddi.1	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵)
3		sbceq1ddi.2	. . . . 5 ⊢ (𝜓 → 𝜃)
4		sbceq1ddi.3	. . . . 5 ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜃)
5	3, 4	sylibr 236	. . . 4 ⊢ (𝜓 → [𝐴 / 𝑥]𝜒)
6	5	adantl 483	. . 3 ⊢ ((𝜑 ∧ 𝜓) → [𝐴 / 𝑥]𝜒)
7	2, 6	sbceq1dd 3731	. 2 ⊢ ((𝜑 ∧ 𝜓) → [𝐵 / 𝑥]𝜒)
8		sbceq1ddi.4	. 2 ⊢ ([𝐵 / 𝑥]𝜒 ↔ 𝜂)
9	7, 8	sylib 220	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜂)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548  [wsbc 3725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-cleq 2733  df-clel 2816  df-sbc 3726

This theorem is referenced by: (None)