Theorem cleq2lem 39975

Description: Equality implies bijection. (Contributed by RP, 24-Jul-2020.)

Hypothesis

Ref	Expression
cleq2lem.b	⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cleq2lem	⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓)))

Proof of Theorem cleq2lem

Step	Hyp	Ref	Expression
1		sseq2 3995	. 2 ⊢ (𝐴 = 𝐵 → (𝑅 ⊆ 𝐴 ↔ 𝑅 ⊆ 𝐵))
2		cleq2lem.b	. 2 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))
3	1, 2	anbi12d 632	1 ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ⊆ wss 3938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-in 3945  df-ss 3954

This theorem is referenced by:  cbvcllem  39976  clublem  39977  rclexi  39982  rtrclex  39984  rtrclexi  39988  clrellem  39989  clcnvlem  39990  trcleq2lemRP  39997  dfrcl2  40026  brtrclfv2  40079  clsk1indlem1  40402